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If 6/(x(x+1))>1, which of the following could the value of x?

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If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 21 Dec 2014, 15:23
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If \(\frac{6}{x(x+1)}>1\), which of the following could the value of x?

A. -3.5
B. -2.5
C. 2.5
D. 3.5
E. 4.5
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 21 Dec 2014, 18:11
1
Hi viktorija,

This question can be solved by TESTing THE ANSWERS. One (and only one) or those numbers could be a solution to the given inequality, so we could check them (just plug them in) until we find one that "fits" the given inequality.

There is a logical math shortcut here though that we can take advantage of:

We're told that 6/(product) > 1 so the denominator must be LESS than 6. That way 6/(less than 6) will be > 1. So we're really just looking for a product that's less than 6. Logically, we're probably looking for a value for X that's relatively close to 0, so let's check answers B and C....But don't do the math just yet...

Answer B: X = -2.5
Denominator = (-2.5)(-1.5)

Answer C: X = 2.5
Denominator = (2.5)(3.5)

Since the negative signs will cancel out in Answer B, you don't have to do the math to see that Answer B is smaller. Since there's only one answer that will "fit", it has to be B.

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 10 Jan 2015, 01:19
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 10 Jan 2015, 03:26
Lucky2783 wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?


Lucky2783 ,I don't think this derivation of yours is right: x(x+1) is +ive implies X is positive. Please check!
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 10 Jan 2015, 05:01
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5


\(\frac{6}{x(x+1)} >1\)

let's analyze the denominator. x(x+1) = x^2+x. now x^2+x will always be positive except for numbers lying between -1 and 0. now if x lies between -1 and 0, then the fraction
\(\frac{6}{x(x+1)}\) will be negative. this violates our initial given condition that \(\frac{6}{x(x+1)} >1\). hence the expression x^2+x will always be positive.

now since expression x^2+x is positive, therefore we can cross multiply. Thus we have
x^2+x<6
x^2+x-6<0
(x+3)(x-2)<0

now for all values of x which are less than -3, (x+3)(x-2) will always be positive. similarly for all values of x , which are greater than 2, (x+3)(x-2) will always be positive.

hence our desired range is between -3 and 2. i.e. -3<x<2. now out of the given options, only option b lies inside this range. hence answer must be B
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 10 Jan 2015, 05:46
sytabish wrote:
Lucky2783 wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?


Lucky2783 ,I don't think this derivation of yours is right: x(x+1) is +ive implies X is positive. Please check!


thanks .
actually we do not need know the sign of X here
Given 6/(x(x+1))>1 implies (x(x+1)) is a +ive quantity so we can simply multilply both sides of inequality by (x(x+1))
6 > (x(x+1))
6> x^2 + x
x^2+x-6 < 0
(x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 10 Jan 2015, 12:37
Hi styabish,

You have to be very careful with your assumptions in the Quant section. This specific Number Property WILL show up on Test Day....

(X)(X+1) = positive

This does NOT mean that X has to be positive.

X COULD be positive....

eg
X = 1
(1)(2) = 2

X COULD be NEGATIVE though...

eg
X = -2
(-2)(-1) = 2

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 12 Jan 2015, 20:56
\(\frac{6}{(x(x+1))} > 1\)

\(\frac{6}{x^2 + x} > 1\)

For x = -3.5

\(\frac{6}{12.25-3.5}\) >> This would be less than 1

For x = -2.5

\(\frac{6}{6.25-2.5}\) >> This would be greater than 1

Answer = B
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 21 Aug 2015, 07:27
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5



isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)
Can someone help, please?
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 21 Aug 2015, 07:36
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5



isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)
Can someone help, please?


Let me ask you: what does x < -3 (x is less than -3) or x < 2 (x is less than 2) even mean?

As for the solution please see above.


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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 21 Aug 2015, 07:41
2
1
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5



isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)
Can someone help, please?


The most straightforward method for this type of question will be to use the values in the options and see which one gives you >1 . Only 1 option must satisfy this requirement.

Additionally, for algebraic solution, look below:

Given : \(\frac{6}{x(x+1)} > 1\) ----> \(\frac{6}{x(x+1)} - 1 > 0\) ---> \(\frac{6-x^2-x}{x(x+1)} > 0\) ----> \(\frac{-6+x^2+x}{x(x+1)} < 0\)

\(\frac{(x+3)(x-2)}{x(x+1)} < 0\) ----> -3<x<-1 or 0<x<2

Only -2.5 lies in this range.

Hope this helps.
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post Updated on: 27 Aug 2015, 01:19
Bunuel wrote:
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5



isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)
Can someone help, please?


Let me ask you: what does x < -3 (x is less than -3) or x < 2 (x is less than 2) even mean?

As for the solution please see above.

This link sorted my worries! For anyone who struggled in the last bit after factorising refer this:
https://www.khanacademy.org/math/algebr ... equalities

Thanks Bunuel! Now I know why did you ask me what didx < -3 (x is less than -3) or x < 2 (x is less than 2) even mean?

Originally posted by aggarwalpooja on 21 Aug 2015, 07:58.
Last edited by aggarwalpooja on 27 Aug 2015, 01:19, edited 1 time in total.
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 21 Aug 2015, 08:15
Engr2012 wrote:
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5



isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)
Can someone help, please?


The most straightforward method for this type of question will be to use the values in the options and see which one gives you >1 . Only 1 option must satisfy this requirement.

Additionally, for algebraic solution, look below:

Given : \(\frac{6}{x(x+1)} > 1\) ----> \(\frac{6}{x(x+1)} - 1 > 0\) ---> \(\frac{6-x^2-x}{x(x+1)} > 0\) ----> \(\frac{-6+x^2+x}{x(x+1)} < 0\)

\(\frac{(x+3)(x-2)}{x(x+1)} < 0\) ----> -3<x<-1 or 0<x<2

Only -2.5 lies in this range.

Hope this helps.


Thanks Engr12, plugging in the value is guess the easiest!
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 02 Jan 2017, 12:43
\(\frac{6}{(x(x+1))} > 1\)

6 > x(x+1)
\(6 > x^{2} + x\)
Plug In options
B
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 20 Feb 2018, 10:10
Lucky2783 wrote:
sytabish wrote:
Lucky2783 wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?


Lucky2783 ,I don't think this derivation of yours is right: x(x+1) is +ive implies X is positive. Please check!


thanks .
actually we do not need know the sign of X here
Given 6/(x(x+1))>1 implies (x(x+1)) is a +ive quantity so we can simply multilply both sides of inequality by (x(x+1))
6 > (x(x+1))
6> x^2 + x
x^2+x-6 < 0
(x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5


hello there! :)
How did you manage to draw the line based on this (x+3)(x-2)<0 is there rule to transform it into line ? :?
thank you :)
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 13 Sep 2018, 15:24
1
Hi dave13,

Let me try to respond to your query.

If \(\frac{6}{x(x+1)}\)>1, which of the following could the value of x?

note x and (x+1) are two consecutive numbers, then their product is always positive . Why
if x is negative then (x+1) is also negative and their product is always positive
if x is positive then (x+1) is also positive and their product is always positive
But x cannot be -1 and 0 , because the exp will be undefined for these values.

So
we can write \(\frac{6}{x(x+1)}\)>1 as \(\frac{6}{x(x+1)}\)-1>0

So \(\frac{(6-x(x+1)}{x(x+1)}\)>0

\(\frac{(6-x^2-x)}{x(x+1)}\)>0

\(\frac{-((x+3)(x-2))}{x(x+1)}\)>0

When we multiply by -1 on both sides we change the sign of inequality.

\(\frac{((x+3)(x-2))}{x(x+1)}\)<0

Now if you draw the number line and have positive and negative regions this is how it would look

++++++++++++(-3)---------------------(-1)++++++++++++(0)--------------------(2)++++++++++

Now the region where the inequality holds is
-3<x<-1 and 0<x<2

Now options B, C, D are greater than 2 so discard.
Option A is less than -3 so discard
Option B lies between -3<x<-1 so this could be possible value of for which the inequality will hold.


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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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New post 14 Sep 2018, 05:09
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5


\(\frac{6}{[x (x + 1)]}\)\(> 1\)

\(\frac{6}{[x^2 + x]}\) > 1

We know that \(x^2 + x\) can never be negative irrespective of the value of \("x"\), therefore

\(6 > x^2 + x\)

Or \(x^2 + x - 6 < 0\)

\((x + 3) (x - 2) < 0\)

\((x - 2) < 0\) or \((x + 3) > 0\)

\(x < 2\) or \(x > -3\)

\(-3 < x < 2\)

Answer : B = \(-2.5\)
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Re: If 6/(x(x+1))>1, which of the following could the value of x? &nbs [#permalink] 14 Sep 2018, 05:09
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